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Theorem andi 837
 Description: Distributive law for conjunction. Theorem *4.4 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
Assertion
Ref Expression
andi ((φ (ψ χ)) ↔ ((φ ψ) (φ χ)))

Proof of Theorem andi
StepHypRef Expression
1 orc 374 . . 3 ((φ ψ) → ((φ ψ) (φ χ)))
2 olc 373 . . 3 ((φ χ) → ((φ ψ) (φ χ)))
31, 2jaodan 760 . 2 ((φ (ψ χ)) → ((φ ψ) (φ χ)))
4 orc 374 . . . 4 (ψ → (ψ χ))
54anim2i 552 . . 3 ((φ ψ) → (φ (ψ χ)))
6 olc 373 . . . 4 (χ → (ψ χ))
76anim2i 552 . . 3 ((φ χ) → (φ (ψ χ)))
85, 7jaoi 368 . 2 (((φ ψ) (φ χ)) → (φ (ψ χ)))
93, 8impbii 180 1 ((φ (ψ χ)) ↔ ((φ ψ) (φ χ)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360 This theorem is referenced by:  andir  838  anddi  840  indi  3501  indifdir  3511  unrab  3526  unipr  3905  uniun  3910  unopab  4638  xpundi  4832  coundir  5083  imadif  5171  unpreima  5408  nmembers1lem3  6270
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